On Bernoulli convolutions generated by second Ostrogradsky series and their fine fractal properties

Abstract

We study properties of Bernoulli convolutions generated by the second Ostrogradsky series, i.e., probability distributions of random variables \begin{equation} \xi = \sum_{k=1}^\infty \frac{(-1)^{k+1}\xi_k}{q_k}, \end{equation} where $q_k$ is a sequence of positive integers with $q_{k+1}\geq q_k(q_k+1)$, and $\{\xi_k\}$ are independent random variables taking the values $0$ and $1$ with probabilities $p_{0k}$ and $p_{1k}$ respectively. We prove that $\xi$ has an anomalously fractal Cantor type singular distribution ($\dim_H (S_{\xi})=0$) whose Fourier-Stieltjes transform does not tend to zero at infinity. We also develop different approaches how to estimate a level of "irregularity" of probability distributions whose spectra are of zero Hausdorff dimension. Using generalizations of the Hausdorff measures and dimensions, fine fractal properties of the probability measure $\mu_\xi$ are studied in details. Conditions for the Hausdorff--Billingsley dimension preservation on the spectrum by its probability distribution function are also obtained.